THE PHILOSOPHY OF LOGICAL ATOMISM

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THE PHILOSOPHY OF LOGICAL ATOMISM.

V. GENERAL PROPOSITIONS AND EXISTENCE.

I AM going to speak to-day about general propositions
and existence. The two subjects really belong to-
gether; they are the same topic, although it might not have
seemed so at the first glance. The propositions and facts
that I have been talking about hitherto have all been such
as involved only perfectly definite particulars, or relations,
or qualities, or things of that sort, never involved the sort
of indefinite things one alludes to by such words as "all,"
"some," "a," "any," and it is propositions and facts of that
sort that I am coming on to to-day.

Really all the propositions of the sort that I mean to
talk of to-day collect themselves into two groups — the first
that are about "all," and the second that are about "some."
These two sorts belong together; they are each other's
negations. If you say, for instance, "All men are mortal,"
that is the negative of "Some men are not mortal." In
regard to general propositions, the distinction of affirma-
tive and negative is arbitrary. Whether you are going to
regard the propositions about "all" as the affirmative ones
and the propositions about "some" as the negative ones,
or vice versa, is purely a matter of taste. For example,
if I say "I met no one as I came along," that, on the face
of it, you would think is a negative proposition. Of course,
that is really a proposition about "all," i. e., "All men are
among those whom I did not meet." If, on the other hand,
I say "I met a man as I came along," that would strike you

THE PHILOSOPHY OF LOGICAL ATOMISM. I9I

as affirmative, whereas it is the negative of "All men are
among those I did not meet as I came along." If you con-
sider such propositions as "All men are mortal" and "Some
men are not mortal," you might say it was more natural
to take the general propositions as the affirmative and the
existence-propositions as the negative, but, simply because
it is quite arbitrary which one is to choose, it is better to for-
get these words and to speak only of general propositions
and propositions asserting existence. All general propo-
sitions deny the existence of something or other. If you
say "All men are mortal," that denies the existence of an
immortal man, and so on.

I want to say emphatically that general propositions
are to be interpreted as not involving existence. When
I say, for instance, "All Greeks are men," I do not want
you to suppose that that implies that there are Greeks. It
is to be considered emphatically as not implying that. That
would have to be added as a separate proposition. If you
want to interpret it in that sense, you will have to add the
further statement "and there are Greeks." That is for
purposes of practical convenience. If you include the fact
that there are Greeks, you are rolling two propositions into
one, and it causes unnecessary confusion in your logic, be-
cause the sorts of propositions that you want are those
that do assert the existence of something and general
propositions which do not assert existence. If it happened
that there were no Greeks, both the proposition that "All
Greeks are men" and the proposition that "No Greeks
are men" would be true. The proposition "No Greeks are
men" is, of course, the proposition "All Greeks are not-
men." Both propositions will be true simultaneously if it
happens that there are no Greeks. All statements about
all the members of a class that has no members are true,
because the contradictory of any general statement does
assert existence and is therefore false in this case. This

192 THE MONIST.

notion, of course, of general propositions not involving
existence is ope which is not in the traditional doctrine of
the syllogism. In the traditional doctrine of the syllogism,
it was assumed that when you have such a statement as
"All Greeks are men," that implies that there are Greeks,
and this produced fallacies. For instance, "All chimeras
are animals, and all chimeras breathe flame, therefore some
animals breathe flame." This is a syllogism in Darapti,
but that mood of the syllogism is fallacious, as this instance
shows. That was a point, by the way, which had a certain
historical interest, because it impeded Leibniz in his at-
tempts to construct a mathematical logic. He was always
engaged in trying to construct such a mathematical logic
as we have now, or rather such a one as Boole constructed,
and he was always failing because of his respect for Aris-
totle. Whenever he invented a really good system, as he
did several times, it always brought out that such moods
as Darapti are fallacious. If you say "All A is B and all
A is C, therefore some B is C" — if you say this you incur
a fallacy, but he could not bring himself to believe that it
was fallacious, so he began again. That shows you that
you should not have too much respect for distinguished
men. 1

Now when you come to ask what really is asserted in
a general proposition, such as "All Greeks are men" for
instance, you find that what is asserted is the truth of all
values of what I call a propositional function. A propo-
sitional function is simply any expression containing an
undetermined constituent, or several undetermined con-
stituents, and becoming a proposition as soon as the un-
determined constituents are determined. If I say "x is a
man" or "n is a number," that is a propositional function ;
so is any formula of algebra, say (x + y) (x — y) =
X s — y 2 . A propositional function is nothing, but, like most

1 Cf. Couturat, La logique de Leibniz.

THE PHILOSOPHY OF LOGICAL ATOMISM. I93

of the things one wants to talk about in logic, it does not
lose its importance through that fact. The only thing
really that you can do with a propositional function is to
assert either that it is always true, or that it is sometimes
true, or that it is never true. If you take :

"If x is a man, x is mortal,"
that is always true (just as much when x is not a man as
when x is a man) ; if you take :

"x is a man,"
that is sometimes true; if you take:

"x is a unicorn,"
that is never true.

One may call a propositional function

necessary, when it is always true;

possible, when it is sometimes true;

impossible, when it is never true.
Much false philosophy has arisen out of confusing
propositional functions and propositions. There is a great
deal in ordinary traditional philosophy which consists
simply in attributing to propositions the predicates which
only apply to propositional functions, and, still worse, some-
times in attributing to individuals predicates which merely
apply to propositional functions. This case of necessary,
possible, impossible, is a case in point. In all traditional
philosophy there comes a heading of "modality," which
discusses necessary, possible, and impossible as properties
of propositions, whereas in fact they are properties of
propositional functions. Propositions are only true or
false.

If you take "x is x," that is a propositional function
which is true whatever "x" may be, i. e., a necessary
propositional function. If you take "x is a man," that is
a possible one. If you take "x is a unicorn," that is an
impossible one.

194 THE MONIST.

Propositions can only be true or false, but propositional
functions have these three possibilities. It is important,
I think, to realize that the whole doctrine of modality only
applies to propositional functions, not to propositions.

Propositional functions are involved in ordinary lan-
guage in a great many cases where one does not usually
realize them. In such a statement as "I met a man," you
can understand my statement perfectly well without know-
ing whom I met, and the actual person is not a constituent
of the proposition. You are really asserting there that a
certain propositional function is sometimes true, namely
the propositional function "I met x and x is human." There
is at least one value of x for which that is true, and that
therefore is a possible propositional function. Whenever
you get such words as "a," "some," "all," "every," it is
always a mark of the presence of a propositional function,
so that these things are not, so to speak, remote or recon-
dite: they are obvious and familiar.

A propositional function comes in again in such a state-
ment as "Socrates is mortal," because "to be mortal" means
"to die at some time or other." You mean there is a time
at which Socrates dies, and that again involves a propo-
sitional function, namely, that "t is a time, and Socrates
dies at t" is possible. If you say "Socrates is immortal,"
that also will involve a propositional function. That means
that "If t is any time whatever, Socrates is alive at time t,"
if we take immortality as involving existence throughout
the whole of the past as well as throughout the whole of
the future. But if we take immortality as only involving
existence throughout the whole of the future, the inter-
pretation of "Socrates is immortal" becomes more com-
plete, viz., "There is a time t, such that if f is any time
later than t, Socrates is alive at f." Thus when you come
to write out properly what one means by a great many
ordinary statements, it turns out a little complicated. "Soc-

THE PHILOSOPHY OF LOGICAL ATOMISM. I95

rates is mortal" and "Socrates is immortal" are not each
other's contradictories, because they both imply that Soc-
rates exists in time, otherwise he would not be either
mortal or immortal. One says, "There is a time at which
he dies," and the other says, "Whatever time you take, he
is alive at that time," whereas the contradictory of "Soc-
rates is mortal" would be true if there is not a time at
which he lives.

An undetermined constituent in a propositional func-
tion is called a variable.

Existence. When you take any propositional function
and assert of it that it is possible, that it is sometimes true,
that gives you the fundamental meaning of "existence."
You may express it by saying that there is at least one
value of x for which that propositional function is true.
Take "x is a man," there is at least one value of x for
which this is true. That is what one means by saying that
"There are men," or that "Men exist." Existence is essen-
tially a property of a propositional function. It means
that that propositional function is true in at least one
instance. If you say "There are unicorns," that will mean
that "There is an x, such that x is a unicorn." That is
written in phrasing which is unduly approximated to ordi-
nary language, but the proper way to put it would be "(x
is a unicorn) is possible." We have got to have some idea
that we do not define, and one takes the idea of "always
true," or of "sometimes true," as one's undefined idea in
this matter, and then you can define the other one as the
negative of that. In some ways it is better to take them
both as undefined, for reasons which I shall not go into at
present. It will be out of this notion of sometimes, which
is the same as the notion of possible, that we get the
notion of existence. To say that unicorns exist is simply
to say that "(x is a unicorn) is possible."

It is perfectly clear that when you say "Unicorns exist,"

I96 THE MONIST.

you are not saying anything that would apply to any uni-
corns there might happen to be, because as a matter of
fact there are not any, and therefore if what you say had
any application to the actual individuals, it could not pos-
sibly be significant unless it were true. You can consider
the proposition "Unicorns exist" and can see that it is
false. It is not nonsense. Of course, if the proposition
went through the general conception of the unicorn to the
individual, it could not be even significant unless there
were unicorns. Therefore when you say "Unicorns exist,"
you are not saying anything about any individual things,
and the same applies when you say "Men exist." If you
say that "Men exist, and Socrates is a man, therefore
Socrates exists," that is exactly the same sort of fallacy
as it would be if you said "Men are numerous, Socrates
is a man, therefore Socrates is numerous," because exist-
ence is a predicate of a propositional function, or deriva-
tively of a class. When you say of a propositional func-
tion that it is numerous, you will mean that there are
several values of x that will satisfy it, that there are more
than one ; or, if you like to take "numerous" in a larger
sense, more than ten, more than twenty, or whatever num-
ber you think fitting. If x, y, and z all satisfy a propo-
sitional function, you may say that that proposition is
numerous, but x, y, and z severally are not numerous.
Exactly the same applies to existence, that is to say that
the actual things that there are in the world do not exist,
or, at least, that is putting it too strongly, because that is
utter nonsense. To say that they do not exist is strictly
nonsense, but to say that they do exist is also strictly non-
sense.

It is of propositional functions that you can assert or
deny existence. You must not run away with the idea
that this entails consequences that it does not entail. If
I say "The things that there are in the world exist," that

THE PHILOSOPHY OF LOGICAL ATOMISM. I97

is a perfectly correct statement, because I am there saying
something about a certain class of things; I say it in the
same sense in which I say "Men exist." But I must not
go on to "This is a thing in the world, and therefore this
exists." It is there the fallacy comes in, and it is simply,
as you see, a fallacy of transferring to the individual that
satisfies a propositional function a predicate which only
applies to a propositional function. You can see this in
A'arious ways. For instance, you sometimes know the
truth of an existence-proposition without knowing any
instance of it. You know that there are people in Tim-
buctoo, but I doubt if any of you could give me an instance
of one. Therefore you clearly can know existence-propo-
sitions without knowing any individual that makes them
true. Existence-propositions do not say anything about
the actual individual but only about the class or function.

It is exceedingly difficult to make this point clear as
long as one adheres to ordinary language, because ordinary
language is rooted in a certain feeling about logic, a cer-
tain feeling that our primeval ancestors had, and as long
as you keep to ordinary language you find it very difficult
to get away from the bias which is imposed upon you by
language. When I say, e. g., "There is an x such that x
is a man," that is not the sort of phrase one would like
to use. "There is an x" is meaningless. What is "an x"
anyhow? There is not such a thing. The only way you
can really state it correctly is by inventing a new language
ad hoc, and making the statement apply straight off to
"x is a man," as when one says "(x is a man) is possible,"
or invent a special symbol for the statement that "x is a
man" is sometimes true.

I have dwelt on this point because it really is of very
fundamental importance. I shall come back to existence
in my next lecture: existence as it applies to descriptions,
which is a slightly more complicated case than I am dis-

I98 THE MONIST.

cussing here. I think an almost unbelievable amount of
false philosophy has arisen through not realizing what
"existence" means.

As I was saying a moment ago, a propositional func-
tion in itself is nothing: it is merely a schema. Therefore
in the inventory of the world, which is what I am trying
to get at, one comes to the question, What is there really
in the world that corresponds with these things ? Of course,
it is clear that we have general propositions, in the same
sense in which we have atomic propositions. For the
moment I will include existence-propositions with general
propositions. We have such propositions as "All men are
mortal" and "Some men are Greeks." But you have not
only such propositions ; you have also such facts, and that,
of course, is where you get back to the inventory of the
world: that, in addition to particular facts, which I have
been talking about in previous lectures, there are also gen-
eral facts and existence-facts, that is to say, there are not
merely propositions of that sort but also facts of that sort.
That is rather an important point to realize. You cannot
ever arrive at a general fact by inference from particular
facts, however numerous. The old plan of complete induc-
tion, which used to occur in books, which was always
supposed to be quite safe and easy as opposed to ordinary
induction, that plan of complete induction, unless it is ac-
companied by at least one general proposition, will not yield
you the result that you want. Suppose, for example, that
you wish to prove in that way that "All men are mortal,"
you are supposed to proceed by complete induction, and
say "A is a man that is mortal," "B is a man that is mortal,"
"C is a man that is mortal," and so on until you finish.
You will not be able, in that way, to arrive at the propo-
sition "All men are mortal" unless you know when you have
finished. That is to say that, in order to arrive by this
road at the general proposition "All men are mortal," you

THE PHILOSOPHY OF LOGICAL ATOMISM. I99

must already have the general proposition "All men are
among those I have enumerated." You never can arrive
at a general proposition by inference from particular prop-
ositions alone. You will always have to have at least one
general proposition in your premises. That illustrates, I
think, various points. One, which is epistemological, is
that if there is, as there seems to be, knowledge of general
propositions, then there must be primitive knowledge of
general propositions (I mean by that, knowledge of general
propositions which is not obtained by inference), because
if you can never infer a general proposition except from
premises of which one at least is general, it is clear that
you can never have knowledge of such propositions by in-
ference unless there is knowledge of some general propo-
sitions which is not by inference. I think that the sort of
way such knowledge — or rather the belief that we have
such knowledge — comes into ordinary life is probably very
odd. I mean to say that we do habitually assume general
propositions which are exceedingly doubtful; as, for in-
stance, one might, if one were counting up the people in
this room, assume that one could see all of them, which is
a general proposition, and very doubtful as there may be
people under the tables. But, apart from that sort of thing,
you do have in any empirical verification of general propo-
sitions some kind of assumption that amounts to this, that
what you do not see is not there. Of course, you would
not put it so strongly as that, but you would assume that,
with certain limitations and certain qualifications, if a thing
does not appear to your senses, it is not there. That is
a general proposition, and it is only through such propo-
sitions that you arrive at the ordinary empirical results
that one obtains in ordinary ways. If you take a census
of the country, for instance, you assume that the people
you do not see are not there, provided you search properly
and carefully, otherwise your census might be wrong. It

200 THE MONIST.

is some assumption of that sort which would underlie what
seems purely empirical. You could not prove empirically
that what you do not perceive is not there, because an
empirical proof would consist in perceiving, and by hypoth-
esis you do not perceive it, so that any proposition of that
sort, if it is accepted, has to be accepted on its own evidence.
I only take that as an illustration. There are many other
illustrations one could take of the sort of propositions that
are commonly assumed, many of them with very little justi-
fication.

I come now to a question which concerns logic more
nearly, namely, the reasons for supposing that there are
general facts as well as general propositions. When we
were discussing molecular propositions I threw doubt upon
the supposition that there are molecular facts, but I do
not think one can doubt that there are general facts. It
is perfectly clear, I think, that when you have enumerated
all the atomic facts in the world, it is a further fact about
the world that those are all the atomic facts there are
about the world, and that is just as much an objective fact
about the world as any of them are. It is clear, I think,
that you must admit general facts as distinct from and
over and above particular facts. The same thing applies
to "AH men are mortal." When you have taken all the
particular men that there are, and found each one of them
severally to be mortal, it is definitely a new fact that all
men are mortal; how new a fact, appears from what I
said a moment ago, that it could not be inferred from
the mortality of the several men that there are in the
world. Of course, it is not so difficult to admit what I
might call existence-facts — such facts as "There are men,"
"There are sheep," and so on. Those, I think, you will
readily admit as separate and distinct facts over and above
the atomic facts I spoke of before. Those facts have got
to come into the inventory of the world, and in that way

THE PHILOSOPHY OF LOGICAL ATOMISM. 201

propositional functions come in as involved in the study
of general facts. I do not profess to know what the right
analysis of general facts is. It is an exceedingly difficult
question, and one which I should very much like to see
studied. I am sure that, although the convenient technical
treatment is by means of propositional functions, that is
not the whole of the right analysis. Beyond that I can-
not go.

There is one point about whether there are molecular
facts. I think I mentioned, when I was saying that I did
not think there were disjunctive facts, that a certain diffi-
culty does arise in regard to general facts. Take "All men
are mortal." That means :

" 'x is a man' implies
'x is a mortal' whatever
x may be."

You can see at once that it is a hypothetical proposition.
It does not imply that there are any men, nor who are
men, and who are not; it simply says that if you have any-
thing which is a man, that thing is mortal. As Mr. Bradley
has pointed out in the second chapter of his Principles of
Logic, "Trespassers will be prosecuted" may be true even
if no one trespasses, since it means merely that, if any one
trespasses, he will be prosecuted. It comes down to this
that

" 'x is a man' implies 'x is a mortal'
is always true,"

is a fact. It is perhaps a little difficult to see how that can
be true if one is going to say that "'Socrates is a man'
implies 'Socrates is a mortal' " is not itself a fact, which
is what I suggested when I was discussing disjunctive
facts. I do not feel sure that you could not get round that
difficulty. I only suggest it as a point which should be
considered when one is denying that there are molecular

202 THE MONIST.

facts, since, if it cannot be got round, we shall have to
admit molecular facts.

Now I want to come to the subject of completely general
propositions and propositional functions. By those I mean
propositions and propositional functions that contain only
variables and nothing else at all. This covers the whole
of logic. Every logical proposition consists wholly and
solely of variables, though it is not true that every propo-
sition consisting wholly and solely of variables is logical.
You can consider stages of generalizations as, e. g.,

"Socrates loves Plato"
"x loves Plato"
"x loves y"
"x R y."

There you have been going through a process of successive
generalization. When you have got to xRy, you have got
a schema consisting only of variables, containing no con-
stants at all, the pure schema of dual relations, and it is
clear that any proposition which expresses a dual relation
can be derived from ^rRy by assigning values to x and R
and y. So that that is, as you might say, the pure form of
all those propositions. I mean by the form of a proposition
that which you get when for every single one of its con-
stituents you substitute a variable. If you want a different
definition of the form of a proposition, you might be in-
clined to define it as the class of all those propositions that
you can obtain from a given one by substituting other
constituents for one or more of the constituents the propo-
sition contains. E. g., in "Socrates loves Plato," you can
substitute somebody else for Socrates, somebody else for
Plato, and some other verb for "loves." In that way there
are a certain number of propositions which you can derive
from the proposition "Socrates loves Plato," by replacing
the constituents of that proposition by other constituents,

THE PHILOSOPHY OF LOGICAL ATOMISM. 203

so that you have there a certain class of propositions, and
those propositions all have a certain form, and one can, if
one likes, say that the form they all have is the class con-
sisting of all of them. That is rather a provisional defini-
tion, because as a matter of fact, the idea of form is more
fundamental than the idea of class. I should not suggest
that as a really good definition, but it will do provisionally
to explain the sort of thing one means by the form of a
proposition. The form of a proposition is that which is
in common between any two propositions of which the one
can be obtained from the other by substituting other con-
stituents for the original ones. When you have got down
to those formulas that contain only variables, like xRy, you
are on the way to the sort of thing that you can assert in
logic.

To give an illustration, you know what I mean by the
domain of a relation : I mean all the terms that have that
relation to something. Suppose I say: "xRy implies that
x belongs to the domain of R," that would be a proposition
of logic and is one that contains only variables. You might
think it contains such words as "belong" and "domain,"
but that is an error. It is only the habit of using ordinary
language that makes those words appear. They are not
really there. That is a proposition of pure logic. It does
not mention any particular thing at all. This is to be
understood as being asserted whatever x and R and y
may be. All the statements of logic are of that sort.

It is not a very easy thing to see what are the con-
stituents of a logical proposition. When one takes "Soc-
rates loves Plato," "Socrates" is a constituent, "loves" is
a constituent, and "Plato" is a constituent. Then you
turn "Socrates" into x, "loves" into R, and "Plato" into y.
x and R and y are nothing, and they are not constituents,
so it seems as though all the propositions of logic were
entirely devoid of constituents. I do not think that can

204 THE MONIST.

quite be true. But then the only other thing you can seem
to say is that the form is a constituent, that propositions
of a certain form are always true: that may be the right
analysis, though I very much doubt whether it is.

There is, however, just this to observe, viz., that the
form of a proposition is never a constituent of that propo-
sition itself. If you assert that "Socrates loves Plato,"
the form of that proposition is the form of the dual rela-
tion, but this is not a constituent of the proposition. If it
were you would have to have that constituent related to the
other constituents. You will make the form much too
substantial if you think of it as really one of the things that
have that form, so that the form of a proposition is cer-
tainly not a constituent of the proposition itself. Never-
theless it may possibly be a constituent of general state-
ments about propositions that have that form, so I think
it is possible that logical propositions might be interpreted
as being about forms.

I can only say, in conclusion, as regards the constituents
of logical propositions, that it is a problem which is rather
new. There has not been much opportunity to consider it.
I do not think any literature exists at all which deals with
it in any way whatever, and it is an interesting problem.

I just want now to give you a few illustrations of propo-
sitions which can be expressed in the language of pure
variables but are not propositions of logic. Among the
propositions that are propositions of logic are included all
the propositions of pure mathematics, all of which cannot
only be expressed in logical terms but can also be deduced
from the premises of logic, and therefore they are logical
propositions. Apart from them there are many that can be
expressed in logical terms, but cannot be proved from logic,
and are certainly not propositions that form part of logic.
Suppose you take such a proposition as : "There is at least
one thing in the world." That is a proposition that you

THE PHILOSOPHY OF LOGICAL ATOMISM. 205

can express in logical terms. It will mean, if you like,
that the propositional function "x =■ x" is a possible one.
That is a proposition, therefore, that you can express in
logical terms ; but you cannot know from logic whether it
is true or false. So far as you do know it, you know it
empirically, because there might happen not to be a uni-
verse, and then it would not be true. It is merely an acci-
dent, so to speak, that there is a universe. The proposition
that there are exactly 30,000 things in the world can also
be expressed in purely logical terms, and is certainly not
a proposition of logic but an empirical proposition (true
or false), because a world containing more than 30,000
things and a world containing fewer than 30,000 things
are both possible, so that if it happens that there are exactly
30,000 things, that is what one might call an accident and
is not a proposition of logic. There are again two propo-
sitions that one is used to in mathematical logic, namely,
the multiplicative axiom and the axiom of infinity. These
also can be expressed in logical terms, but cannot be proved
or disproved by logic. In regard to the axiom of infinity,
the impossibility of logical proof or disproof may be taken
as certain, but in the case of the multiplicative axiom, it
is perhaps still open to some degree to doubt. Everything
that is a proposition of logic has got to be in some sense
or other like a tautology. It has got to be something that
has some peculiar quality, which I do not know how to
define, that belongs to logical propositions and not to others.
Examples of typical logical propositions are:

"If p implies q and q implies r, then
p implies r."

"If all a's are b's and all &'s are c's,
then all a's are c's.

"If all a's are b's, and x is an a, then
x is a &."

206 THE MONIST.

Those are propositions of logic. They have a certain
peculiar quality which marks them out from other propo-
sitions and enables us to know them a priori. But what
exactly that characteristic is, I am not able to tell you.
Although it is a necessary characteristic of logical propo-
sitions that they should consist solely of variables, i. e.,
that they should assert the universal truth, or the some-
times-truth, of a propositional function consisting wholly
of variables — although that is a necessary characteristic,
it is not a sufficient one.

I am sorry that I have had to leave so many problems
unsolved. I always have to make this apology, but the
world really is rather puzzling and I cannot help it.

DISCUSSION.

Is there any word you would substitute for "existence"

which would give existence to individuals ? Are you applying
the word "existence" to two ideas, or do you deny that there
are two ideas?

Mr. Russell: No, there is not an idea that will apply to individuals.
As regards the actual things there are in the world, there is
nothing at all you can say about them that in any way cor-
responds to this notion of existence. It is a sheer mistake to
say that there is anything analogous to existence that you
can say about them. You get into confusion through lan-
guage, because it is a perfectly correct thing to say "All the
things in the world exist," and it is so easy to pass from
this to "This exists because it is a thing in the world." There
is no sort of point in a predicate which could not conceivably
be false. I mean, it is perfectly clear that, if there were such
a thing as this existence of individuals that we talk of, it
would be absolutely impossible for it not to apply, and that
is the characteristic of a mistake.

VI. DESCRIPTIONS AND INCOMPLETE SYMBOLS.

I am proposing to deal this time with the subject of
descriptions, and what I call "incomplete symbols," and

THE PHILOSOPHY OF LOGICAL ATOMISM. 207

the existence of described individuals. You will remember
that last time I dealt with the existence of kinds of things,
what you mean by saying "There are men" or "There are
Greeks" or phrases of that sort, where you have an exist-
ence which may be plural. I am going to deal to-day with
an existence which is asserted to be singular, such as "The
man with the iron mask existed" or some phrase of that
sort, where you have some object described by the phrase
"The so-and-so" in the singular, and I want to discuss the
analysis of propositions in which phrases of that kind
occur.

There are, of course, a great many propositions very
familiar in metaphysics which are of that sort: "I exist"
or "God exists" or "Homer existed," and other such state-
ments are always occurring in metaphysical discussions,
and are, I think, treated in ordinary metaphysics in a way
which embodies a simple logical mistake that we shall be
concerned with to-day, the same sort of mistake that I
spoke of last week in connection with the existence of kinds
of things. One way of examining a proposition of that
sort is to ask yourself what would happen if it were false.
If you take such a proposition as "Romulus existed," prob-
ably most of us think that Romulus did not exist. It is
obviously a perfectly significant statement, whether true
or false, to say that Romulus existed. If Romulus himself
entered into our statement, it would be plain that the state-
ment that he did not exist would be nonsense, because you
cannot have a constituent of a propositon which is nothing
at all. Every constituent has got to be there as one of
the things in the world, and therefore if Romulus himself
entered into the propositions that he existed or that he did
not exist, both these propositions could not only not be true,
but could not be even significant, unless he existed. That
is obviously not the case, and the first conclusion one draws
is that, although it looks as if Romulus were a constituent

208 THE MONIST.

of that proposition, that is really a mistake. Romulus does
not occur in the proposition "Romulus did not exist."

Suppose you try to make out what you do mean by that
proposition. You can take, say, all the things that Livy
has to say about Romulus, all the properties he ascribes
to him, including the only one probably that most of us
remember, namely, the fact that he was called "Romulus."
You can put all this together, and make a propositional
function saying "x has such-and-such properties," the prop-
erties being those you find enumerated in Livy. There
you have a propositional function, and when you say that
Romulus did not exist you are simply saying that that
propositional function is never true, that it is impossible
in the sense I was explaining last time, i. e., that there is
no value of x that makes it true. That reduces the non-
existence of Romulus to the sort of non-existence I spoke
of last time, where we had the non-existence of unicorns.
But it is not a complete account of this kind of existence
or non-existence, because there is one other way in which
a described individual can fail to exist, and that is where
the description applies to more than one person. You can-
not, e. g., speak of "The inhabitant of London," not because
there are none, but because there are so many.

You see, therefore, that this proposition "Romulus ex-
isted" or "Romulus did not exist" does introduce a propo-
sitional function, because the name "Romulus" is not really
a name but a sort of truncated description. It stands for
a person who did such-and-such things, who killed Remus,
and founded Rome, and so on. It is short for that descrip-
tion ; if you like, it is short for "the person who was called
'Romulus/ " If it were really a name, the question of
existence could not arise, because a name has got to name
something or it is not a name, and if there is no such person
as Romulus there cannot be a name for that person who
is not there, so that this single word "Romulus" is really

THE PHILOSOPHY OF LOGICAL ATOMISM. 209

a sort of truncated or telescoped description, and if you
think of it as a name you will get into logical errors. When
you realize that it is a description, you realize therefore
that any proposition about Romulus, really introduces the
propositional function embodying the description, as (Say)
"x was called 'Romulus.' " That introduces you at once
to a propositional function, and when you say "Romulus
did not exist," you mean that this propositional function
is not true for one value of x.

There are two sorts of descriptions, what one may call
"ambiguous descriptions," when we speak of "o so-and-so,"
.and what one may call "definite descriptions," when we
speak of "the so-and-so" (in the singular). Instances are:

Ambiguous: A man, a dog, a pig, a Cabinet Minister.

Definite : The man with the iron mask.

The last person who came into this room.
The only Englishman who ever occupied the

Papal See.
The number of the inhabitants of London.
The sum of 43 and 34.

(It is not necessary for a description that it should describe
an individual: it may describe a predicate or a relation or
anything else.)

It is phrases of that sort, definite descriptions, that I
want to talk about to-day. I do not want to talk about
ambiguous descriptions, as what there was to say about
them was said last time.

I want you to realize that the question whether a phrase
is a definite description turns only upon its form, not upon
the question whether there is a definite individual so de-
scribed. For instance, I should call "The inhabitant of
London" a definite description, although it does not in fact
describe any definite individual.

The first thing to realize about a definite description

210 THE MONIST.

is that it is not a name. We will take "The author of
Waverley." That is a definite description, and it is easy
to see that it is not a name. A name is a simple symbol
(i. e., a symbol which does not have any parts that are
symbols), a simple symbol used to designate a certain
particular or by extension an object which is not a par-
ticular but is treated for the moment as if it were, or is
falsely believed to be a particular, such as a person. This
sort of phrase, "The author of Waverley," is not a name
because it is a complex symbol. It contains parts which
are symbols. It contains four words, and the meanings
of those four words are already fixed and they have fixed
the meaning of "The author of Waverley" in the only
sense in which that phrase does have any meaning. In
that sense, its meaning is already determinate, i. e., there
is nothing arbitrary or conventional about the meaning
of that whole phrase, when the meanings of "the,"
"author," "of," and "Waverley" have already been fixed.
In that respect, it differs from "Scott," because when you
have fixed the meaning of all the other words in the lan-
guage, you have done nothing toward fixing the meaning
of the name "Scott." That is to say, if you understand
the English language, you would understand the meaning
of the phrase "The author of Waverley" if you had never
heard it before, whereas you would not understand the
meaning of "Scott" if you had never heard the word be-
fore because to know the meaning of a name is to know
who it is applied to.

You sometimes find people speaking as if descriptive
phrases were names, and you will find it suggested, e. g.,
that such a proposition as "Scott is the author of Waverley"
really asserts that "Scott" and "the author of Waverley"
are two names for the same person. That is an entire
delusion; first of all, because "the author of Waverley" is
not a name, and, secondly, because, as you can perfectly

THE PHILOSOPHY OF LOGICAL ATOMISM. 211

well see, if that were what is meant, the proposition would
be one like "Scott is Sir Walter," and would not depend
upon any fact except that the person in question was so
called, because a name is what a man is called. As a
matter of fact, Scott was the author of Waverley at a time
when no one called him so, when no one knew whether he
was or not, and the fact that he was the author was a
physical fact, the fact that he sat down and wrote it with
his own hand, which does not have anything to do with
what he was called. It is in no way arbitrary. You can-
not settle by any choice of nomenclature whether he is or
is not to be the author of Waverley, because in actual fact
he chose to write it and you cannot help yourself. That
illustrates how "the author of Waverley" is quite a dif-
ferent thing from a name. You can prove this point very
clearly by formal arguments. In "Scott is the author of
Waverley" the "is," of course, expresses identity, i. e.,
the entity whose name is Scott is identical with the author
of Waverley. But, when I say "Scott is mortal" this "is"
is the "is" of predication, which is quite different from the
"is" of identity. It is a mistake to interpret "Scott is
mortal" as meaning "Scott is identical with one among
mortals," because (among other reasons) you will not be
able to say what "mortals" are except by means of the
propositional function "x is mortal," which brings back
the "is" of predication. You cannot reduce the "is" of
predication to the other "is." But the "is" in "Scott is the
author of Waverley" is the "is" of identity and not of
predication. 1

If you were to try to substitute for "the author of
Waverley" in that proposition any name whatever, say
"c" so that the proposition becomes "Scott is c" then if V
is a name for anybody who is not Scott, that proposition

* The confusion of these two meanings of "is" is essential to the Hegelian
conception of identity-in-diflference.

212 THE MONIST.

would become false, while if, on the other hand, "c" is a
name for Scott, then the proposition will become simply
a tautology. It is at once obvious that if "c" were "Scott"
itself, "Scott is Scott" is just a tautology. But if you
take any other name which is just a name for Scott, then
if the name is being used as a name and not as a description,
the proposition will still be a tautology. For the name itself
is merely a means of pointing to the thing, and does not
occur in what you are asserting, so that if one thing has
two names, you make exactly the same assertion whichever
of the two names you use, provided they are really names
and not truncated descriptions.

So there are only two alternatives. If "c" is a name,
the proposition "Scott is c" is either false or tautologous.
But the proposition "Scott is the author of Waverley" is
neither, and therefore is not the same as any proposition
of the form "Scott is c" where "c" is a name. That is
another way of illustrating the fact that a description is
quite a different thing from a name.

I should like to make clear what I was saying just now,
that if you substitute another name in place of "Scott"
which is also a name of the same individual, say, "Scott
is Sir Walter," then "Scott" and "Sir Walter" are being
used as names and not as descriptions, your proposition is
strictly a tautology. If one asserts "Scott is Sir Walter,"
the way one would mean it would be that one was using the
names as descriptions. One would mean that the person
called "Scott" is the person called "Sir Walter," and "the
person called 'Scott' " is a description, and so is "the per-
son called 'Sir Walter.' " So that would not be a tautol-
ogy. It would mean that the person called "Scott" is
identical with the person called "Sir Walter." But if you
are using both as names, the matter is quite different.
You must observe that the name does not occur in that
which you assert when you use the name. The name is

THE PHILOSOPHY OF LOGICAL ATOMISM. 213

merely that which is a means of expressing what it is you
are trying to assert, and when I say "Scott wrote Waver-
ley," the name "Scott" does not occur in the thing I am
asserting. The thing I am asserting is about the person,
not about the name. So if I say "Scott is Sir Walter,"
using these two names as names, neither "Scott" nor "Sir
Walter" occurs in what I am asserting, but only the person
who has these names, and thus what I am asserting is a
pure tautology.

It is rather important to realize this about the two
different uses of names or of any other symbols: the one
when you are talking about the symbol and the other when
you are using it as a symbol, as a means of talking about
something else. Normally, if you talk about your dinner,
you are not talking about the word "dinner" but about
what you are going to eat, and that is a different thing
altogether. The ordinary use of words is as a means of
getting through to things, and when you are using words
in that way the statement "Scott is Sir Walter" is a pure
tautology, exactly on the same level as "Scott is Scott."

That brings me back to the point that when you take
"Scott is the author, of Waverley" and you substitute for
"the author of Waverley" a name in the place of a descrip-
tion, you get necessarily either a tautology or a falsehood —
a tautology if you substitute "Scott" or some other name
for the same person, and a falsehood if you substitute any-
thing else. But the proposition itself is neither a tautology
nor a falsehood, and that shows you that the proposition
"Scott is the author of Waverley" is a different proposi-
tion from any that can be obtained if you substitute a
name in the place of "the author of Waverley." That
conclusion is equally true of any other proposition in which
the phrase "the author of Waverley" occurs. If you take
any proposition in which that phrase occurs and substitute
for that phrase a proper name, whether that name be

214 THE MONIST.

"Scott" or any other, you will get a different proposition.
Generally speaking, if the name that you substitute is
"Scott," your proposition, if it was true before will remain
true, and if it was false before will remain false. But it
is a different proposition. It is not always true that it will
remain true or false, as may be seen by the example:
"George IV wished to know if Scott was the author of
Waverley." It is not true that George IV wished to know
if Scott was Scott. So it is even the case that the truth
or the falsehood of a proposition is sometimes changed
when you substitute a name of an object for a description
of the same object. But in any case it is always a different
proposition when you substitute a name for a description.

Identity is a rather puzzling thing at first sight. When
you say "Scott is the author of Waverley," you are half-
tempted to think there are two people, one of whom is Scott
and the other the author of Waverley, and they happen to
be the same. That is obviously absurd, but that is the sort
of way one is always tempted to deal with identity.

When I say "Scott is the author of Waverley" and that
"is" expresses identity, the reason that identity can be
asserted there truly and without tautology is just the fact
that the one is a name and the other a description. Or
they might both be descriptions. If I say "The author
of Waverley is the author of Marmion," that, of course,
asserts identity between two descriptions.

Now the next point that I want to make clear is that
when a description (when I say "description" I mean, for
the future, a definite description) occurs in a proposition,
there is no constituent of that proposition corresponding
to that description as a whole. In the true analysis of the
proposition, the description is broken up and disappears.
That is to say, when I say "Scott is the author of Waver-
ley" it is a wrong analysis of that to suppose that you
have there three constituents, "Scott," "is," and "the author

THE PHILOSOPHY OF LOGICAL ATOMISM. 215

of Waverley." That, of course, is the sort of way you
might think of analyzing. You might admit that "the
author of Waverley" was complex and could be further
cut up, but you might think the proposition could be split
into those three bits to begin with. That is an entire
mistake. "The author of Waverley" is not a constituent
of the proposition at all. There is no constituent really
there corresponding to the descriptive phrase. I will try
to prove that to you now.

The first and most obvious reason is that you can have
significant propositions denying the existence of "the so-
and-so." "The unicorn does not exist." "The greatest
finite number does not exist." Propositions of that sort
are perfectly significant, are perfectly sober, true, decent
propositions, and that could not possibly be the case if the
unicorn were a constituent of the proposition, because
plainly it could not be a constituent as long as there were
not any unicorns. Because the constituents of proposi-
tions, of course, are the same as the constituents of the
corresponding facts, and since it is a fact that the unicorn
does not exist, it is perfectly clear that the unicorn is not
a constituent of that fact, because if there were any fact
of which the unicorn was a constituent, there would be a
unicorn, and it would not be true that it did not exist. That
applies in this case of descriptions particularly. Now since
it is possible for "the so-and-so" not to exist and yet for
propositions in which "the so-and-so" occurs to be sig-
nificant and even true, we must try to see what is meant
by saying that the so-and-so does exist.

The occurrence of tense in verbs is an exceedingly
annoying vulgarity due to our preoccupation with practical
affairs. It would be much more agreeable if they had no
tense, as I believe is the case in Chinese, but I do not know
Chinese. You ought to be able to say "Socrates exists in
the past," "Socrates exists in the present" or "Socrates

2l6 THE MONIST.

exists in the future," or simply "Socrates exists," without
any implication of tense, but language does not allow that,
unfortunately. Nevertheless, I am going to use language
in this tenseless way: when I say "The so-and-so exists,"
I am not going to mean that it exists in the present or in
the past or in the future, but simply that it exists, without
implying anything involving tense.

"The author of Waverley exists" : there are two things
required for that. First of all, what is "the author of
Waverley" ? It is the person who wrote Waverley, i. e., we
are coming now to this, that you have a propositional
function involved, viz., "x writes Waverley" and the author
of Waverley is the person who writes Waverley, and in
order that the person who writes Waverley may exist, it
is necessary that this propositional function should have
two properties:

i. It must be true for at least one x.

2. It must be true for at most one x.

If nobody had ever written Waverley the author could not
exist, and if two people had written it, the author could
not exist. So that you want these two properties, the one
that it is true for at least one x, and the other that it is
true for at most one x, both of which are required for ex-
istence.

The property of being true for at least one x is the one
we dealt with last time: what I expressed by saying that
the propositional function is possible. Then we come on
to the second condition, that it is true for at most one x,
and that you can express in this way: "If x and y wrote
Waverley, then x is identical with y, whatever x and y
may be." That says that at most one wrote it. It does
not say that anybody wrote Waverley at all, because if
nobody had written it, that statement would still be true.
It only says that at most one person wrote it.

THE PHILOSOPHY OF LOGICAL ATOMISM. 21"J

The first of these conditions for existence fails in the
case of the unicorn, and the second in the case of the in-
habitant of London.

We can put these two conditions together and get a
portmanteau expression including the meaning of both.
You can reduce them both down to this, that: "('x wrote
Waverley' is equivalent to 'x is c' whatever x may be) is
possible in respect of c." That is as simple, I think, as you.
can make the statement.

You see that means to say that there is some entity c,
we may not know what it is, which is such that when x
is c, it is true that x wrote Waverley, and when x is not c,
it is not true that x wrote Waverley, which amounts to
saying that c is the only person who wrote Waverley ; and
I say there is a value of c which makes that true. So
that this whole expression, which is a propositional func-
tion about c, is possible in respect of c (in the sense ex-
plained last time).

That is what I mean when I say that the author of
Waverley exists. When I say "the author of Waverley
exists," I mean that there is an entity c such that "x wrote
Waverley" is true when x is c, and is false when x is not c.
"The author of Waverley" as a constituent has quite dis-
appeared there, so that when I say "The author of Waver-
ley exists" I am not saying anything about the author of
Waverley. You have instead this elaborate to-do with
propositional functions, and "the author of Waverley" has
disappeared. That is why it is possible to say significantly
"The author of Waverley did not exist." It would not be
possible if "the author of Waverley" were a constituent
of propositions in whose verbal expression this descriptive
phrase occurs.

The fact that you can discuss the proposition "God
exists" is a proof that "God," as used in that proposition,

2l8 THE MONIST.

is a description and not a name. If "God" were a name,
no question as to existence could arise.

I have now defined what I mean by saying that a thing
described exists. I have still to explain what I mean by
saying that a thing described has a certain property. Sup-
posing you want to say "The author of Waverley was
human," that will be represented thus: "('x wrote Waver-
ley' is equivalent to 'x is c' whatever x may be, and c is
human) is possible with respect to c."

You will observe that what we gave before as the
meaning of "The author of Waverley exists" is part of
this proposition. It is part of any proposition in which
"the author of Waverley" has what I call a "primary oc-
currence." When I speak of a "primary occurrence I
mean that you are not having a proposition about the
author of Waverley occurring as a part of some larger
proposition, such as "I believe that the author of Waverley
was human" or "I believe that the author of Waverley
exists." When it is a primary occurrence, i. e., when
the proposition concerning it is not just part of a larger
proposition, the phrase which we defined as the meaning
of "The author of Waverley exists" will be part of that
proposition. If I say the author of Waverley was human,
or a poet, or a Scotsman, or whatever I say about the
author of Waverley in the way of a primary occurrence,
always this statement of his existence is part of the propo-
sition. In that sense all these propositions that I make
about the author of Waverley imply that the author of
Waverley exists. So that any statement in which a de-
scription has a primary occurrence implies that the object
described exists. If I say "The present King of France
is bald," that implies that the present King of France
exists. If I say, "The present King of France has a fine
head of hair," that also implies that the present King of
France exists. Therefore unless you understand how a

THE PHILOSOPHY OF LOGICAL ATOMISM. 210.

proposition containing a description is to be denied, you
will come to the conclusion that it is not true either that the
present King of France is bald or that he is not bald,
because if you were to enumerate all the things that are
bald you would not find him there, and if you were to
enumerate all the things that are not bald, you would not
find him there either. The only suggestion I have found
for dealing with that on conventional lines is to suppose
that he wears a wig. You can only avoid the hypothesis
that he wears a wig by observing that the denial of the
proposition "The present King of France is bald" will not
be "The present King of France is not bald," if you mean
by that "There is such a person as the King of France and
that person is not bald." The reason of this is that when
you state that the present King of France is bald you say
"There is a c such that c is now King of France and c is
bald" and the denial is not "There is a c such that c is
now King of France and c is not bald." It is more com-
plicated. It is : "Either there is not a c such that c is now
King of France, or, if there is such a c, then c is not bald."
Therefore you see that, if you want to deny the proposition
"The present King of France is bald," you can do it by
denying that he exists, instead of by denying that he is
bald. In order to deny this statement that the present
King of France is bald, which is a statement consisting
of two parts, you can proceed by denying either part.
You can deny the one part, which would lead you to sup-
pose that the present King of France exists but is not bald,
or the other part, which will lead you to the denial that the
present King of France exists; and either of those two
denials will lead you to the falsehood of the proposition
"The present King of France is bald." When you say
"Scott is human" there is no possibility of a double denial.
The only way you can deny "Scott is human" is by saying

220 THE MONIST.

"Scott is not human." But where a descriptive phrase
occurs, you do have the double possibility of denial.

It is of the utmost importance to realize that "the so-
and-so" does not occur in the analysis of propositions in
whose verbal expression it occurs, that when I say "The
author of Waverley is human," "the author of Waverley"
is not the subject of that proposition, in the sort of way
that Scott would be if I said "Scott is human," using "Scott"
as a name. I cannot emphasize sufficiently how important
this point is, and how much error you get into metaphysics
if you do not realize that when I say "The author of
Waverley is human" that is not a proposition of the same
form as "Scott is human." It does not contain a con-
stituent "the author of Waverley." The importance of
that is very great for many reasons, and one of them is
this question of existence. As I pointed out to you last
time, there is a vast amount of philosophy that rests upon
the notion that existence is, so to speak, a property that
you can attribute to things, and that the things that exist
have the property of existence and the things that do not
exist do not. That is rubbish, whether you take kinds of
things, or individual things described. When I say, e. g.,
"Homer existed," I am meaning by "Homer" some de-
scription, say "the author of the Homeric poems," and I
am asserting that those poems were written by one man,
which is a very doubtful proposition ; but if you could get
hold of the actual person who did actually write those
poems (supposing there was such a person), to say of him
that he existed would be uttering nonsense, not a falsehood
but nonsense, because it is only of persons described that it
can be significantly said that they exist. Last time I pointed
out the fallacy in saying "Men exist, Socrates is a man, there-
fore Socrates exists." When I say "Homer exists, this is
Homer, therefore this exists," that is a fallacy of the same
sort. It is an entire mistake to argue: "This is the author

THE PHILOSOPHY OF LOGICAL ATOMISM. 221

of the Homeric poems and the author of the Homeric
poems exists, therefore this exists." It is only where a
propositional function comes in that existence may be sig-
nificantly asserted. You can assert "The so-and-so exists,"
meaning that there is just one c which has those properties,
but when you get hold of a c that has them, you cannot
say of this c that it exists, because that is nonsense: it is
not false, but it has no meaning at all.

So the individuals that there are in the world do not
exist, or rather it is nonsense to say that they exist and
nonsense to say that they do not exist. It is not a thing
you can say when you have named them, but only when you
have described them. When you say "Homer exists," you
mean "Homer" is a description which applies to something.
A description when it is fully stated is always of the form
"the so-and-so."

The sort of things that are like these descriptions in
that they occur in words in a proposition, but are not in
actual fact constituents of the proposition rightly analyzed,
things of that sort I call "incomplete symbols." There are
a great many sorts of incomplete symbols in logic, and they
are sources of a great deal of confusion and false philos-
ophy, because people get misled by grammar. You think
that the proposition "Scott is mortal" and the proposition
"The author of Waverley is mortal" are of the same form.
You think that they are both simple propositions attributing
a predicate to a subject. That is an entire delusion: one
of them is (or rather might be) and one of them is not.
These things, like "the author of Waverley," which I call
incomplete symbols, are things that have absolutely no
meaning whatsoever in isolation but merely acquire a mean-
ing in a context. "Scott" taken as a name has a meaning
all by itself. It stands for a certain person, and there it is.
But "the author of Waverley" is not a name, and does not
all by itself mean anything at all, because when it is rightly

222 THE MONIST.

used in propositions, those propositions do not contain any
constituent corresponding to it.

There are a great many other sorts of incomplete sym-
bols besides descriptions. These are classes, which I shall
speak of next time, and relations taken in extension, and
so on. Such aggregations of symbols are really the same
thing as what I call "logical fictions," and they embrace
practically all the familiar objects of daily life: tables,
chairs, Piccadilly, Socrates, and so on. Most of them are
either classes, or series, or series of classes. In any case
they are all incomplete symbols, i. e, they are aggregations
that only have a meaning in use and do not have any
meaning in themselves.

It is important, if you want to understand the analysis
of the world, or the analysis of facts, or if you want to
have any idea what there really is in the world, to realize
how much of what there is in phraseology is of the nature
of incomplete symbols. You can see that very easily in the
case of "the author of Waverley" because "the author of
Waverley" does not stand simply for Scott, nor for any-
thing else. If it stood for Scott, "Scott is the author of
Waverley" would be the same proposition as "Scott is
Scott," which it is not, since George IV wished to know
the truth of the one and did not wish to know the truth
of the other. If "the author of Waverley" stood for any-
thing other than Scott, "Scott is the author of Waverley"
would be false, which it is not. Hence you have to conclude
that "the author of Waverley" does not, in isolation, really
stand for anything at all; and that is the characteristic of
incomplete symbols.

[to be concluded.]

Bertrand Russell.
London, England.